If all the possible outcomes of an event are equally likely, then the theoretical probability of an event, $E$E, happening is given by:
$P(E)=\frac{\text{number of favourable outcomes }}{\text{total number of outcomes }}$P(E)=number of favourable outcomes total number of outcomes
For example, say we wanted to know the probability of rolling a $3$3 on a die. There are six possible outcomes in total (the sample space is {$1,2,3,4,5,6$1,2,3,4,5,6}). However, there is only one three on a die.
So, I could write $P(3)=\frac{1}{6}$P(3)=16
If an experiment has equally likely outcomes, it means the chance of any outcome occurring is equal. In other words, there is no bias. Bias is a favouring of one outcome over all others.
Worked example
A bag of marbles contains $4$4 red marbles, $5$5 green marbles and $3$3 black marbles. A marble is randomly drawn from the bag. Find:
a) P(red)
Think: There are $4$4 red marbles out of a total of $12$12 marbles.
Do: P(red) = $\frac{4}{12}$412 = $\frac{1}{3}$13
b) P(red or black)
Think: There are $4$4 red and $3$3 black marbles out of a total of $12$12 marbles. We add the number of red and black together and write as a fraction
Do: P(red or black) = $\frac{\text{4 + 3}}{12}$4 + 312 = $\frac{7}{12}$712
c) P(red) + P(green)
Think: We need to find the probabilities separately then add them together.
Do: P(red) =$\frac{4}{12}$412 and P(green) = $\frac{5}{12}$512. P(red) + P(green) =$\frac{4}{12}+\frac{5}{12}=\frac{7}{12}$412+512=712
Use the following applet to practise calculating the theoretical probability.
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Investigation: Relative frequency vs theoretical probability
Relative frequency gives us an estimate or the probability of an event that is based on performing trials of an experiment.
Theoretical probability is the expected probability of an event calculated by using the formula stated above.
When we toss a coin $20$20 times, we would expect heads to come up half the time, that is $10$10 times. However, when we actually perform the experiment the results may not reflect that.
Go to this site http://www.shodor.org/interactivate/activities/Coin/ select "show cumulative stats" and select "ratio" to display the results.
Compare the number of times heads occur when you flip the coin $20$20 times, $100$100 times, $1000$1000 times. Comment on your findings. What do you think the results will look like if we flipped the coin an infinite number of times?
Practice questions
Question 1
A number is randomly selected from the following list:
$\left\{1,3,3,6,6,6,8,8,8,8,10,10,10,10,10\right\}${1,3,3,6,6,6,8,8,8,8,10,10,10,10,10}
How many numbers are in this list?
$\editable{}$
What is the probability of selecting a $1$1?
What is the probability of selecting a $3$3?
What is the probability of selecting an $8$8?
Which number is most likely to be selected?
$\editable{}$
Question 2
From a normal deck of cards, what is the probability of selecting:
a two?
A four?
Not a seven?
A red card?
A fifteen?
A face card?
question 3
A marble is randomly drawn from a bag which contains 6 red marbles, 7 green marbles and 3 blue marbles. Find:
P(red)+P(green)+P(blue)
P(red or green)
- P(red or blue)
- P(green or blue)